Question: $ A = \left[\begin{array}{rr}-2 & 2 \\ 2 & 5\end{array}\right]$ $ C = \left[\begin{array}{rrr}0 & 1 & 5 \\ 5 & 3 & 3\end{array}\right]$ What is $ A C$ ?
Because $ A$ has dimensions $(2\times2)$ and $ C$ has dimensions $(2\times3)$ , the answer matrix will have dimensions $(2\times3)$ $ A C = \left[\begin{array}{rr}{-2} & {2} \\ {2} & {5}\end{array}\right] \left[\begin{array}{rrr}{0} & \color{#DF0030}{1} & \color{#9D38BD}{5} \\ {5} & \color{#DF0030}{3} & \color{#9D38BD}{3}\end{array}\right] = \left[\begin{array}{rrr}? & ? & ? \\ ? & ? & ?\end{array}\right] $ To find the element at any row $i$ , column $j$ of the answer matrix, multiply the elements in row $i$ of the first matrix, $ A$ , with the corresponding elements in column $j$ of the second matrix, $ C$ , and add the products together. So, to find the element at row 1, column 1 of the answer matrix, multiply the first element in ${\text{row }1}$ of $ A$ with the first element in ${\text{column }1}$ of $ C$ , then multiply the second element in ${\text{row }1}$ of $ A$ with the second element in ${\text{column }1}$ of $ C$ , and so on. Add the products together. $ \left[\begin{array}{rrr}{-2}\cdot{0}+{2}\cdot{5} & ? & ? \\ ? & ? & ?\end{array}\right] $ Likewise, to find the element at row 2, column 1 of the answer matrix, multiply the elements in ${\text{row }2}$ of $ A$ with the corresponding elements in ${\text{column }1}$ of $ C$ and add the products together. $ \left[\begin{array}{rrr}{-2}\cdot{0}+{2}\cdot{5} & ? & ? \\ {2}\cdot{0}+{5}\cdot{5} & ? & ?\end{array}\right] $ Likewise, to find the element at row 1, column 2 of the answer matrix, multiply the elements in ${\text{row }1}$ of $ A$ with the corresponding elements in $\color{#DF0030}{\text{column }2}$ of $ C$ and add the products together. $ \left[\begin{array}{rrr}{-2}\cdot{0}+{2}\cdot{5} & {-2}\cdot\color{#DF0030}{1}+{2}\cdot\color{#DF0030}{3} & ? \\ {2}\cdot{0}+{5}\cdot{5} & ? & ?\end{array}\right] $ Fill out the rest: $ \left[\begin{array}{rrr}{-2}\cdot{0}+{2}\cdot{5} & {-2}\cdot\color{#DF0030}{1}+{2}\cdot\color{#DF0030}{3} & {-2}\cdot\color{#9D38BD}{5}+{2}\cdot\color{#9D38BD}{3} \\ {2}\cdot{0}+{5}\cdot{5} & {2}\cdot\color{#DF0030}{1}+{5}\cdot\color{#DF0030}{3} & {2}\cdot\color{#9D38BD}{5}+{5}\cdot\color{#9D38BD}{3}\end{array}\right] $ After simplifying, we end up with: $ \left[\begin{array}{rrr}10 & 4 & -4 \\ 25 & 17 & 25\end{array}\right] $